I've been looking at pairs of positive integers which have integer power means, including the harmonic, geometric, arithmetic, quadratic (RMS), cubic, and quartic means. Conjecture 1: No two positive integers have a cubic mean that's an integer. Conjecture 2: No two positive integers have a quartic mean that's an integer. Conjecture 3: No two positive integers have both a geometric mean that's an integer and a quadratic mean that's an integer. All three are true for all possible pairs of positive integers less than ten thousand. Are they true for all larger integers too? Other than that, all possible combinations appear to exist. For instance 5 and 45 have a harmonic mean of 9, a geometric mean of 15, and an arithmetic mean of 25. 4 and 28 have a harmonic mean of 7, an arithmetic mean of 16, and a quadratic mean of 20. And plenty of pairs have just 0, 1, or 2 means that are integers, in all possible combinations that don't violate my conjectures. Does anyone have a proof or counterexample for any of my conjectures? Thanks.